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\title{MATH0037 Galois Theory 2002, Sheet 5}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
\begin{document}
\maketitle
\begin{enumerate}
\item Suppose that 
$f = \sum_{i=0}^N a_i X^i \in \mathbb Z[X]$ (with $N >0$) and 
that $p$ is a prime number with the following properties.
\begin{enumerate} 
\item $p \mid a_i$ for all $i > 0$,
\item $p \not\ \mid a_0$ and
\item $p^2 \not\ \mid a_N$,
\end{enumerate}
By considering the ring epimorphism $\overline \pi : \mathbb Z[X]
\rightarrow \mathbb Z_p[X]$ defined by taking the coefficients
modulo $p$, show that $f$ is not factorizable as a product 
of two integral polynomials of smaller degree. Deduce that
$f$ is irreducible as a rational polynomial by appeal to
a result of Gauss.
\item Let $F$ be a field and suppose that
$c \in F$. Consider maps $\alpha_c, \beta : F[X]
\rightarrow F[X]$ defined as follows. In each case
$0 \not = f = \sum_{i=0}^N a_i X^i$ has degree $N \geq 0$
and is otherwise arbitrary.
\begin{enumerate}
\item $\alpha_c : f \mapsto \sum_{i=0}^N a_i (X+c)^i$  and $0 \mapsto 0$.
\item $\beta : f \mapsto X^N f(1/X)$ and $0 \mapsto 0$.
\end{enumerate}
Show that each of these maps is an {\em automorphism} of the 
ring $F[X]$ (i.e. an isomorphism from $F[X]$ with itself) and
that each map preserves degree. Deduce that these automorphisms,
and their inverses, send irreducible elements of $F[X]$ to irreducible
elements of $F[X]$.

\item Which of the following polynomials are irreducible when viewed as
rational polynomials?
\begin{enumerate}
\item $14X^7 + 7X + 2$.
\item $(\sum_{i=0}^N (i+1)! x^i) + 42X^{N+1}$ where $N \geq 0$.
\item $2X^7 + 7X^6 + 14$. {\em Hint. Use the previous question. The map
$\beta$ may be useful}. 
\item $X^2 + X + 1$. {\em Hint. Use the previous question. The map
$\alpha_1$ may be useful}.
\item $X^3 + X^2 + X + 1$.
\item $X^4 + X^3 + X^2 + X + 1$.
\item $X^5 + X^4 + X^3 + X^2 + X + 1$.
\item $X^6 + X^5 + X^4 + X^3 + X^2 + X + 1$.
\item $X^{p-1} + X^{p-2} + \cdots + X + 1$ where $p$ is a prime number.
\end{enumerate} 
\item Suppose that 
$f = \sum_{i=0}^N a_i X^i \in \mathbb Z[X]$ (with $N >0$) and
that $p$ is a prime number with the following properties.
\begin{enumerate}
\item $p \mid a_i$ for all $i < N$,
\item $p \not \ \mid a_N$ and
\item $p^2 \not \ \mid a_0$.
\end{enumerate}
Prove that $f$ is irreducible when viewed as a rational polynomial.
\end{enumerate}
\end{document}
